Russell, Ch.33 of Principles of Mathematics, ‘Real Numbers

by Corry Shores  

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Bertrand Russell

Principles of Mathematics

Part 5: Infinity and Continuity

Ch.33: The Real Numbers

Brief Summary:

Rational numbers are ‘compact’, meaning that between any two there is another. A class of rationals can have a determinate upper limit, like those not greater than r [thus having r as the upper limit], or they will not have such a limit, for example those less than r [and thus the upper figure gets closer and closer to r without having a final determinate value before it.] The second kind, those without upper bounds, are called segments.

Summary

§258

Real numbers are opposed to rational numbers, in fact they are really not numbers at all.  [272]

Real numbers are comprised of the rational and irrational numbers, and the irrational numbers are defined as the limits of such series of rationals as have neither a rational nor an infinite limit. But Russell says this definition is problematic, and there may not be any such irrationals as defined above.

§259

In the series of rational numbers, there is always another number between any two. “The rational numbers in order of magnitude form a series in which there is a term between any two.” [273] Because Cantor uses the term ‘continuous’ for another purpose, Russell will use the term ‘compact’ for this trait of rationals [it might be equivalent to ‘dense’].

in a compact series, there are an infinite number of terms between any two, there are no consecutive terms and the stretch between any two terms (whether these be included or not) is again a compact series.
[273]

Russell now distinguishes 4 infinite classes for rationals. consider any rational r. Regarding classes of other rationals, there are:

(1) those less than r,

(2) those not greater than r [this is the same as (1), except it includes r]

(3) those greater than r, and

(4) those not less than r [similarly, (4) contains r]

[273]

But another difference between the pairings is that (2) has a last term, but (1) does not [because between r and any near it, no matter how near, there will always be another.] Also, “(1) is identical with the class of rational numbers less than a variable term of (1), while (2) does not have this characteristic.” [273] Classes with the properties if (1) are called ‘segments’.

A segment of rationals may be defined as a class of rationals which is not null, nor yet coextensive with the rationals themselves (i.e. which contains some but not all rationals), and which is identical with the class of rationals less than a (variable) term of itself, i.e. with the class of rationals x such that there is a rational y of the said class such that x is less than y.
[273]

After showing how to find a segment from finite or infinite classes of rationals, u, he concludes:

If u be a single rational, or a class of rationals all of which are less than some fixed rational, then the rationals less than u, if u be a single term, or less than a variable term of u, if u be a class of terms, always form a segment of rationals. My contention is, that a segment of rationals is a real number.
[274]

§260

segments are not capable of a one-one correlation with rationals. There are classes of rationals defined as being composed of all terms | less than a variable term of an infinite class of rationals, which are not definable as all the rationals less than some one definite rational. Moreover there are more segments than rationals, and hence the series of segments has continuity of a higher order than the rationals. Segments form a series in virtue of the relation of whole and part, or of logical inclusion (excluding identity). Any two segments are such that one of them is wholly contained in the other, and in virtue of this fact they form a series. It can be easily shown that they form a compact series. What is more remarkable is this: if we apply the above process to the series of segments, forming segments of segments by reference to classes of segments, we find that every segment of segments can be defined as all segments contained in a certain definite segment. Thus the segment of segments defined by a class of segments is always identical with the segment of segments defined by some one segment. Also every segment defines a segment of segments which can be defined by an infinite class of segments. These two properties render the series of segments perfect, in Cantor’s language; but the explanation of this term must be left till we come to the doctrine of limits.
[275]

  

§261

A given segment may be defined by many different classes of rationals. Two such classes u and v may be regarded as having the segment as a common property. Two infinite classes u and v will define the same lower segment if, given any u, there is a v greater than it, and given any v, there is a u greater than it. If each class has no maximum, this is also a necessary condition. The classes u and v are then what Cantor calls coherent (zusammengehörig).

[…]

We have now seen that the usual properties of real numbers belong to segments of rationals. There is therefore no mathematical reason for distinguishing such segments from real numbers. It remains to set forth, first the nature of a limit, then the current theories of irrationals, and then the objections which make the above theory seem preferable.
[276]

Bertrand Russell. Principles of Mathematics. London/New York: Routledge, 2010 [1st published 1903].